The extent of the imperial Chinese bureaucracy under the T'ang dynasty makes possible an unusually thorough scientific project (echoing, for a different purpose, the brave amateur experiment of Eratosthenes 1000 years earlier). In 721 the emperor sets up nine research stations, across a span of more than 2000 miles, from Hue in the south to the Great Wall in the north.

For four years each station measures the sun's shadow at noon on the summer and winter solstice. It is an elegant experiment in that no difficult synchronization is required. The shortest and longest shadows at each place are the correct answers, providing invaluable information for cartographers.

A famous map of 801 - a landmark in cartography - no doubt makes use of the nine points of latitude scientifically established in the experiment of 721-5. It is a map of the Chinese world, produced for the T'ang emperor by Chia Tan.

Chia Tan's map is on an ambitious scale, measuring about 10 by 11 yards. It charts the entire T'ang empire and extends its range into the barbarian world beyond China's borders, showing the seven main trade routes with other parts of Asia.

Mappa Mundi: 13th century

The medieval form of map known simply as Mappa Mundi ('map of the world') features in many manuscripts, particularly in the 13th century. Of about 1000 surviving versions some are small illustrations in bound volumes; others are impressively large. The Mappa Mundi kept in Hereford cathedral measures 64' by 54'.

All follow the same cartographic principle, based on Christian rather than geographical considerations.

In the convention of a Mappa Mundi the world is shown as a flat circle (even though well known to be a sphere). At the centre is Jerusalem; at the top, in the place of honour, is paradise presided over by Jesus. From this the alignment of the map follows in an inevitable hierarchy.

The holiest part of the map is with Jesus at the top; and the holiest point of the compass is the east (the position of the altar in a church). The eastern continent, Asia, must therefore occupy the top half of the map. Jerusalem, the site of the Crucifixion in west Asia, falls conveniently in the exact centre. In a straight line due west to the bottom of the map, as if suspended from Jerusalem, is the Mediterranean.

With the Mediterranean as a vertical dividing line in the lower half of the circle, the position of the other two known continents is fixed. Europe becomes the bottom left quarter, Africa the bottom right.

They are neatly separated from Asia by two waterways, dividing the circle horizontally. To the left of centre, the Black Sea and the river Don; to the right of centre, the Nile. So the waterways form a neat T-shape in the bottom half of the surrounding O, in this most schematic of map conventions. Dragons and monsters conveniently fill up the spaces in which there are no known geographical features.

Portolan charts: 14th century

The experience of practical navigators begins to prevail in maps produced from about 1300, showing the Mediterranean, the Black Sea and the Atlantic coast. Known as portolan charts, they contain more accurate detail than any previous maps. They are criss-crossed by lines joining the main harbours.

These lines tell sailors what course to set on their recently introduced navigational aid, the magnetic compass. With the accuracy made possible by the compass, feedback from the navigators in turn improves the charts. But this scientific approach to mapmaking is about to suffer a setback, with the European rediscovery of Ptolemy.

Ptolemy and the Renaissance: 15th century

In the gradual spread of ancient Greek texts to medieval Europe, manuscripts of Ptolemy become known by 1400. His account of world geography is widely available after it is translated into Latin in 1410.

With the arrival of printing later in the century, a world map based on Ptolemy's information is a natural project for the publishers. The first printed version, done from engraved copper plates, appears in Bologna in 1477. The projection of the map is redrawn and made clearer in the 1482 German edition, printed in Ulm from wood blocks.

The Ptolemaic map shows the known world, from the Atlantic coast in the west to China and India in the east. India stretches on through what we now call Indonesia, to reach the edge of the map below China. The supposed ocean separating Europe from China and India is the unseen region behind the map.

Luckily for the story of exploration, this ocean is assumed to be relatively small (Ptolemy greatly reduces the more accurate figure for the circumference of the earth arrived at by Eratosthenes). The unseen ocean is small on the world's first globe. And it is on this same assumption that Columbus sails west, just fifteen years after the first printed edition of Ptolemy, confident that he will soon reach the coast of India or China.

Another Ptolemaic error is disproved by the explorers just a few years later. Even though Herodotus reported that a Phoenician fleet had sailed round the southern tip of Africa, the Ptolemaic map shows south Africa extending east through terra incognita to join up with India in the far East, making the Indian Ocean a vast inland sea.

In 1497 Vasco da Gama makes his way round the Cape of Good Hope, pioneering the sea route to India which he reaches in 1498. Rarely until the 20th century has new technology, in this case the printed map, been so rapidly outdated.

The first globe: 1492

One of the most unfortunate innovators in the history of invention is Martin Behaim, the creator of the world's first globe - made in Nuremberg in 1492.

His idea is excellent. A globe is the only accurate way of representing the surface of the earth. His misfortune is to base his globe on Ptolemy (who postulates a single ocean between Spain and China) and to achieve his three-dimensional version of this notion in the very year in which it is disproved - by Columbus reaching America. But Behaim shows the reason for Columbus's confidence in sailing west. The distance on his globe between Spain and China is only half what it should be.